Question

Please help me with this question!

Design a new cereal box that will hold the same amount of cereal but reduce manufacturing costs. Prove that your new design holds the same amount but can be manufactured more cheaply.

The original box was a rectangular prism and had the surface area of 334^2 and a volume of 312in^2

Answer 1

now.. notice the picture below

it has a front-and-back of 11x4

it has a left-and-right of 6x4

and a top-and-bottom of 6x11

now, if the Surface Area is less than 290in², then it is cheaper to manufacture because it uses less material to make the box

explanation 2 If you dont understand Explanation 1

The new cereal box is in the form of a cube.

So from the formula of volume of cube a a³

Where a is the side of the cube.

This volume is equal to the volume of the original box

a³ = 264 inch³

a = 6.415 inch

Now the surface area of the box will be

S = 6a²

S = 6 × (6.415)² = 246.91 inch²

Which is less than the surface area of the original box.

Therefore, the manufacturing cost of the new cereal box will be less.

Explanation:

Answer 2

Here is my proposed redesign of the cereal box to reduce manufacturing costs while maintaining the same volume:

Original box dimensions:

- Surface Area: 334 in^2

- Volume: 312 in^3

Since surface area is proportional to the cost of materials and manufacturing, a design with lower surface area will cost less to produce.

Proposed new design: Cylinder

To hold the same 312 in^3 volume, the cylinder would have:

Radius: 3.68 inches

Height: 8.46 inches

Surface Area of cylinder:

2πrh + 2πr^2

= 2(3.14)(3.68)(8.46) + 2(3.14)(3.68)^2

= 300 in^2

Since the surface area of the cylindrical box is less than the original at 300 in^2 versus 334 in^2, the cylindrical design will require less material and cost less to produce while maintaining the same 312 in^3 volume to hold the same amount of cereal.

In summary, by changing from a rectangular prism shape to a cylindrical shape with a radius of 3.68 inches and height of 8.46 inches, we can reduce the surface area from 334 in^2 to 300 in^2. This lower surface area translates to lower material costs and manufacturing costs while still providing the original 312 in^3 volume capacity.